Nonlinear Wave Motion

波浪运动是自然界的基础：光波影响人们的视觉，声波使人们能够听到，而水波对船舶和潮汐能的运动至关重要。 该项目的重点是非线性波传播，即大振幅或高功率信号的研究，旨在提高科学家的能力，以发展对大振幅波和波能转移的相关动力学的洞察力和理解。研究结果将有助于非线性波动的应用，包括海啸，流氓或畸形波，高功率激光，以及压力波的大变化的传播，如冲击波和爆炸波中发生的那些。对于线性波或小振幅波，有一个实质性的理论，在许多情况下，对于非线性波，现有的理论发展较少，已知精确解的非线性波方程相对较少。然而，所谓的逆散射变换方法已被用来找到解决方案，并了解基本方程的重要属性，并导致了广泛的一类物理上重要的非线性波动方程的解决方案。这种方法可以用来找到所谓的孤子解，稳定的局域波，具有特殊的相互作用特性，并已被发现广泛应用，例如在水波，电磁波，晶格动力学，磁波和弹性。研究人员将考虑扩展这种方法的新类的非线性方程，如分数阶非线性波动方程及其孤子解。分数方程本身有许多应用，包括对无定形材料的研究。此外，该方法将被应用于研究一类多维孤子的性质，即在所有方向上消失的块型波，这也与量子力学和光学中出现的方程有关。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Wave motion is fundamental in nature: Light waves impact the way people see, sound waves allow people to hear, and water waves are critical to the motion of ships and tidal energy. This project is focused on nonlinear wave propagation, that is the study of large amplitude or high-power signals, and aims to enhance the ability of scientists to develop insight and understanding of large amplitude waves and of the associated dynamics of wave energy transfer. The results of the investigation will benefit applications of nonlinear wave motion, which include tsunamis, rogue or freak waves, high power lasers, and the propagation of large changes in pressure waves such as those that occur in shock waves and blast waves.For linear waves or small amplitude waves a substantial theory is available and, in many situations, exact solutions can be found. For nonlinear waves the available theory is less developed and there are relatively few nonlinear wave equations for which exact solutions are known. However, the so-called Inverse Scattering Transform method has been used to find solutions and understand important properties of the underlying equations and has led to a wide class of solutions of physically significant nonlinear wave equations. This method can be used to find so-called solitons solutions, stable localized waves which have special interaction properties and have been found to apply widely, for example in water waves, electromagnetic waves, lattice dynamics, magnetic waves, and elasticity. The investigator will consider extensions of this method to new classes of nonlinear equations, such as fractional nonlinear wave equations and their soliton solutions. Fractional equations themselves have numerous applications, including to the study of amorphous materials. Additionally, the method will be applied to study the properties of a class of multidimensional solitons, that is lump-type waves that vanish in all directions, which are relevant also for equations that arise in quantum mechanics and optics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.